Note that, unlike sensitivity, the dimension is less dependent on the exact type of distance function. For example, using Euclidean distance or Euclidean distance to the power of 3 as a variant for the \(k\) -means clustering problem does not change the dimension. This is because set of ranges for both of these problems is the same: subsets of the input points that can be covered by \(k\)  balls. It is easy to compute the dimension for the \(k\) -means problem (the query space \((P,w,Q,g)\)  in our paper, as well as the modified query space for the function \(f\) . These bounds can be found in [1]}. In addition,  [1]} provide a simple reduction that shows that the dimension for \(k\)  centers is the same as the dimension of 1 center multiplied by \(k\) .
